To find the de Broglie wavelength of an object, we can use the formula derived from quantum mechanics, which relates the wavelength to the momentum of the object. The de Broglie wavelength (λ) is given by the equation:
Formula for de Broglie Wavelength
λ = h / p
Where:
- λ is the de Broglie wavelength.
- h is Planck's constant (approximately 6.626 × 10-34 Js).
- p is the momentum of the object, calculated as p = mv, where m is mass and v is velocity.
Calculating the Wavelengths
Now, let's calculate the de Broglie wavelength for each of the objects you mentioned:
(a) Bullet of mass 0.040 kg traveling at 1.0 km/s
First, we need to convert the speed from km/s to m/s:
1.0 km/s = 1000 m/s
Now, we calculate the momentum:
p = mv = 0.040 kg × 1000 m/s = 40 kg·m/s
Next, we can find the de Broglie wavelength:
λ = h / p = (6.626 × 10-34 Js) / (40 kg·m/s) = 1.6565 × 10-35 m
(b) Ball of mass 0.060 kg moving at 1.0 m/s
Here, we calculate the momentum:
p = mv = 0.060 kg × 1.0 m/s = 0.060 kg·m/s
Now, we find the de Broglie wavelength:
λ = h / p = (6.626 × 10-34 Js) / (0.060 kg·m/s) = 1.1043 × 10-32 m
(c) Dust particle of mass 1.0 × 10-9 kg drifting at 2.2 m/s
Calculating the momentum for the dust particle:
p = mv = (1.0 × 10-9 kg) × (2.2 m/s) = 2.2 × 10-9 kg·m/s
Now, we can determine the de Broglie wavelength:
λ = h / p = (6.626 × 10-34 Js) / (2.2 × 10-9 kg·m/s) = 3.0082 × 10-25 m
Summary of Results
- Bullet: λ ≈ 1.6565 × 10-35 m
- Ball: λ ≈ 1.1043 × 10-32 m
- Dust particle: λ ≈ 3.0082 × 10-25 m
These calculations illustrate how the de Broglie wavelength decreases as the mass and speed of the object increase, highlighting the wave-particle duality of matter. In everyday life, the wavelengths of macroscopic objects like bullets and balls are so small that they are not observable, while microscopic particles like dust can have relatively larger wavelengths in comparison, though still extremely tiny. This concept is fundamental in quantum mechanics and helps explain the behavior of particles at very small scales.
